Alysane Mormont, perhaps? She has 2 children and is unmarried. She tells Asha that she "turned into a bear and found a mate in the woods". My tinfoil hat is telling me their father may be Tormund Giantboner!

It is! I just read the passage so it stuck out to me. It's on pg 555, during Asha's first chapter in ADWD (titled 'The King's Prize').

"Aye." Alysane stared at Asha for a moment. "I have a son. He's only two. My daughter's nine."

"You started young."

"Too young. But better that than wait too late."

A stab at me, Asha thought, but let it be. "You are wed."

"No. My children were fathered by a bear." Alysane smiled. Her teeth were crooked, but there was something ingratiating about that smile. "Mormont women are skinchangers. We turn into bears and find mates in the woods. Everyone knows."

I'm not taking the skinchanger part literally, but instead perhaps it was a rumor circulated for certain Mormont women to hide the fact that they took wildling lovers.

After thinking about it, my tinfoil is pretty convinced that Alysane Mormont's two illegitimate children are in fact Tormund's little bear cubs.

There was a terrible prophesy concerning it, that has been all but lost in the mists of time. All that remains of the prophesy are three words, carried by an ancient and honourable house. But the true meaning has been lost; and has been replaced with a far more prosaic warning. His member has a name, and that name is Winter.

Let me lay it in the snow, it brings nothing but woe. But that that junk let him bunk with the all manner of hoes
On a mammoth big and hairy he patrolled beyond the wall, with wildlings and giants at his beck and call.
There goes Tormund, bane of the giants! His giant member however keeps them nice and pliant. Ate the chickens of kangs and bawdy songs he sang; he's coming, he's cumming, he's coming!
TORMUND....TORMUND! 12 foot cock, fucking killing for fun! Spread spread many legs; he's coming, he's cumming, he's coming!
Throw me in a black cell if I go too fast; but its a fact John Snow wishes he was his dad! Got a cock for your mom and an axe for your dad; the crows think he's bad but he's a pretty cool lad. OOOO, he'll save children but not Crasters children. He'll screw woman but not Crasters woman. He'll might screw children but not CRASTERS children!
(Jujujujujujujujujuju)
Stole a whole bunch of woman, fucked the SHIT out of bears, was nursed by a giant, and kills crows with a stare. He makes love like a kraken sliding into the sea. Clears a whole hill of snow when he has to make wee.
Tormund...Tormund
Twelve store cock set for fornication
The Dormish beware, Kingslanding beware! He's coming, he's cumming, he's coming!

TL;DR
I MADE A POORLY WRITTEN SONG FOR TORMUND GIANTS BANE USING THE THEME AND SONG "Washington" BY BRAD NEELY